The given problem involves differentiating the function $g(x) = e^{x^2 - x}$, using the properties of exponential functions and basic differentiation rules.

Step-by-step solution:

1. Function definition:

$g(x) = e^{x^2 - x}$

2. Differentiation:

We need to apply the chain rule to differentiate this function. The chain rule states that: $\frac{d}{dx} e^{u(x)} = e^{u(x)} \cdot u'(x)$ where $u(x) = x^2 - x$.

• First, differentiate $u(x) = x^2 - x$ with respect to $x$: $u'(x) = 2x - 1$

• Now, apply the chain rule to $g(x)$: $g'(x) = e^{x^2 - x} \cdot (2x - 1)$

Final expression:

Thus, the derivative of $g(x)$ is: $g'(x) = (2x - 1) \cdot e^{x^2 - x}$

Correct alternative:

Looking at the provided alternatives, option b) matches this derivative: $g'(x) = (2x - 1) \cdot e^{x^2 - x}$

Would you like further details or have any questions?

Here are 5 related questions to deepen understanding:

1. How does the chain rule apply to other exponential functions like $e^{\sin(x)}$?
2. What is the difference between differentiating polynomial and exponential functions?
3. Can you find the second derivative of $g(x)$?
4. How would the process differ if the function were $g(x) = \ln(x^2 - x)$?
5. How does the derivative of exponential functions differ from trigonometric functions?

Tip:

When applying the chain rule, always first identify the inner function and its derivative before multiplying by the derivative of the outer function.